Solve the inequality LaTeX: \displaystyle \frac{8}{x^{2} - 25}<\frac{5}{x^{2} + x - 30}

Getting zero on one side and factoring gives LaTeX: \displaystyle - \frac{5}{\left(x - 5\right) \left(x + 6\right)} + \frac{8}{\left(x - 5\right) \left(x + 5\right)}< 0 . This gives the least common denominator as LaTeX: \displaystyle \left(x - 5\right) \left(x + 5\right) \left(x + 6\right) . Building each fraction to get the common denominator gives LaTeX: \displaystyle \frac{8 x + 48 - (5 x + 25)}{\left(x - 5\right) \left(x + 5\right) \left(x + 6\right)} < 0 . Simplifying gives LaTeX: \displaystyle \frac{3 x + 23}{\left(x - 5\right) \left(x + 5\right) \left(x + 6\right)}<0 . The inequality can change signs at the zeros of the numerator, LaTeX: \displaystyle \left\{- \frac{23}{3}\right\} , or the zeros of the denominator LaTeX: \displaystyle \left\{-6, -5, 5\right\} . Making a sign chart gives: This gives the solution LaTeX: \displaystyle \left(- \frac{23}{3}, -6\right) \cup \left(-5, 5\right)